Proving that the totient function increases infinitely

[two lock]

Homework Statement

Prove that $\varphi$(n)→∞ as n→∞. n$\in$Z

Homework Equations

$\varphi$(n) = the number of integers less than n that are coprime to n.

The Attempt at a Solution

My professor said that we need to show that $\varphi$(n) is always greater than some increasing estimate of itself. I'm really not sure how to even go about applying this. Any help you could offer would be really appreciated.

What I was thinking of doing would be to say let k be the integer with the maximum value of the totient function. k=p₁^$\alpha$₁...p_n^$\alpha$_n where p_i is a prime number. So $\varphi$(k) = $\varphi$(p₁^$\alpha$₁...p_n^$\alpha$_n)=$\varphi$(p₁^$\alpha$₁)...$\varphi$(p_n^$\alpha$_n). Multiply k by any prime greater than p_n, called p_l. Call this number j. Then $\varphi$(j)=$\varphi$(p₁^$\alpha$₁)...$\varphi$(p_n^$\alpha$_n)$\varphi$(p_l^$\alpha$_l). This new multiple is greater than zero (I would write out the basis case in order to guarantee this), so the second equation is greater than the first, so there cannot be an integer with the maximum value of the totient. In other words, as n increases, the totient function increases.

 

[lippyka]

I'm not sure if this is really what my professor meant by "show that \varphi(n) is always greater than some increasing estimate of itself." If anyone could provide insight into what this means and how I should go about proving it, I would really appreciate it.